Two-Dimensional Patterns With Distinct Differences—Constructions, Bounds, and Maximal Anticodes

Abstract

Abstract-A two-dimensional grid with dots is called a configuration with distinct differences if any two lines which connect two dots are distinct either in their length or in their slope. These configurations are known to have many applications such as radar, sonar, physical alignment, and time-position synchronization. Rather than restricting dots to lie in a square or rectangle, as previously studied, we restrict the maximum distance between dots of the configuration; the motivation for this is a new application of such configurations to key distribution in wireless sensor networks. We consider configurations in the hexagonal grid as well as in the traditional square grid, with distances measured both in the Euclidean metric, and in the Manhattan or hexagonal metrics.We note that these configurations are confined inside maximal anticodes in the corresponding grid. We classify maximal anticodes for each diameter in each grid. We present upper bounds on the number of dots in a pattern with distinct differences contained in these maximal anticodes. Our bounds settle (in the negative) a question of Golomb and Taylor on the existence of honeycomb arrays of arbitrarily large size. We present constructions and lower bounds on the number of dots in configurations with distinct differences contained in various two-dimensional shapes (such as anticodes) by considering periodic configurations with distinct differences in the square grid.

Sequence Folding, Lattice Tiling, and Multidimensional Coding

“…The two-dimensional problems has also interest from discrete geometry point of view and it was discussed for example in [11], [12]. Recent new application in keys predistribution for wireless sensor networks [13] led to new related twodimensional problems concerning these patterns which are discussed in [14], [15]. It has raised the following discrete geometry problem: given a regular polygon with area s on the square (or hexagonal) grid, what is the maximum number of grid points that can be taken, such that any two lines connecting these grid points are different either in their length or in their slope.…”

“…It has raised the following discrete geometry problem: given a regular polygon with area s on the square (or hexagonal) grid, what is the maximum number of grid points that can be taken, such that any two lines connecting these grid points are different either in their length or in their slope. Upper bound technique based on an idea of Erdös and Turán [11], [16] is given in [14]. Some preliminary lower bounds on the number of dots are also given in [14], where the use of folding is applied.…”

“…Upper bound technique based on an idea of Erdös and Turán [11], [16] is given in [14]. Some preliminary lower bounds on the number of dots are also given in [14], where the use of folding is applied. Folding for such patterns was first used by [10].…”

“…The generalization to higher dimensions is straight forward. F3 was used in [14] to obtain some synchronization patterns in Z D .…”

Sidon sequences and doubly periodic two-dimensional synchronization patterns

“…Several types of two-dimensional synchronization arrays are defined in the literature. We start we a general definition which was given in [3], [5]. Let S be a given shape, on the square grid, with m dots on grid points.…”